By natural I mean that we could have simple laws of physics and initial
conditions in which the creatures evolve over a long period of time,
as we have seen in our universe.

It is very likely that even Conway's Life universe has this feature. Its
rules are absurdly simple, and we know that it can contain self-replicating
structures, which would be capable of mutation, and therefore evolution. We
can specify very simple initial conditions from which self-replicating
structures would be overwhelmingly likely to appear, as long as the lattice
is big enough. (The binary digits of many easily-computable real numbers
would work.) Moving from this 2D world, in which each cell can be pictured
as a square with 4 orthogonal neighbors, we can consider 3D CA in which
each cell is a cube with 6 orthogonal neighbors. There are rule sets and
initial conditions for this lattice structure that are just as simple as
Conway's life, which can similarly contain evolving self-replicating
structures. We can go further and envision a 4D CA in which each cell is a
hypercube with 8 orthogonal neighbors. Without a doubt, there are absurdly
simple rulesets for this lattice structure which are computation universal,
support stable structures like gliders, and support self-replicating
structures which would grow and evolve.

Universes of the natural type would seem likely to have higher measure,
because they are inherently simpler to specify.

If that's true, then the CA universes described above should have very high
measure, because they are extremely simple to specify.

Tegmark goes into some detail on the
problems with other than 3+1 dimensional space.

Once again, I don't see how these problems apply to 4D CA. His arguments
are extremely physics-centric ones having to do with what happens when you
tweak quantum-mechanical or string-theory models of our particular universe.